a) An even function is such that f(x)=f(-x).
ln|(1-e⁻ˣ)/(1+e⁻ˣ)|=
ln|(eˣ-1)/(eˣ+1)|=
ln|(1-eˣ)/(1+eˣ)|.
Therefore if f(x)=ln|(1-e⁻ˣ)/(1+e⁻ˣ)|, f(x) is even: TRUE
b) Differentiate:
3x²+12ydy/dx+5=0.
At (-1,1) [which satisfies the equation of the curve, so the point is on the curve], we get: 3+12dy/dx+5=0, dy/dx=-⅔. This is perpendicular to neither axis: so FALSE.
c) d/dx{∫[1,√x](tan²(t²)dt)}=tan²(x)/(√x-1), where the integral is between a low limit of 1 and a high limit of √x and x∈[1,∞). If the equation were true for all values of x in the given range, then it has to be true for x=1. But that would make the right-hand side expression infinite, so it’s FALSE.
Also, d(F(t²))/dt=2tF'(t), that is, ∫2tF'(t)dt=F(t²), and applying the interval we would have F(x)-F(1), which differentiates to F'(x), because F(1) is a constant. But the term 2t is missing from the integrand which is just tan²(t²) and not 2ttan²(t²).
d) For x<0, y=0 and for x≥0, y=2x, where x∈ℚ, so y is only strictly increasing when x≥0; therefore FALSE.
e) f(x)=|x| is integrable for all x but not differentiable at x=0 because its gradient is not definable at x=0. Compare with f(x)=x which is integrable and everywhere differentiable (gradient=1). Integration can be defined as the area under a curve (or straight line for linear functions) and the area under f(x)=|x| is defined because it’s a continuous function, but it’s not differentiable everywhere. So: FALSE.
